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Chapter 2 Describing Contingency Tables Reported by Liu Qi

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Review of Chapter 1 Categorical variable Response-Explanatory variable Nominal-Ordinal-Interval variable Continuous-Discrete variable Quantitative-Qualitative variable

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Review(cont.) Use binomial, multinomial and Poisson distribution Not normality distribution Tow most used models: logistic regression(logit) log linear

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Binomial distribution

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Multinomial distribution

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Poisson distribution

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Poisson Multinomial

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Something unfamiliar Maximum likelihood estimation Confidence intervals Statistical inference for binomial parameters multinomial parameters ……

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Terminology and notation Cell Contingency table

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Terminology and notation Subjective Sensitivity and Specificity Conditional distribution Joint distribution Marginal distribution Independence =>

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Sampling Scheme Poisson the joint probability mass function: Multinomial independent/product multinomial Hyper geometric

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Example for sampling

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Types of studies Retrospective: case-control Prospective: – Clinical trial observational study – Cohort study Cross-sectional: experimental study

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Comparing two proportions Difference Relative risk Odds ratio – Odds defined as – For a 2*2 table, odds ratio – Another name: cross-product ratio

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Properties of the Odds Ratio 0=<θ <, θ=1 means independence of X and Y the farther from 1.0, the stronger the association between X and Y. log θ is convenient and symmetric Suitable for all direction No change when any row/column multiplied by a constant.

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Aspirin and Heart Attacks Revisited 189/11034=0.0171 104/11037=0.0094 Relative risk: 0.0171/0.0094=1.82 Odds ratio: (189*10933)/(10845*1 04)=1.83

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Case-Control Studies and the Odds Ratio

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However(cont.)

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Partial association in stratified 2*2 tables Experimental studies We hold other covariates constant to study the effect of X on Y. Observational studies Control for a possibly confounding variable Z Partial tables=>conditional association Marginal table

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Death penalty example

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Death penalty example(cont.)

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Simpsons paradox

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Conditional and marginal odds ratios Conditional Marginal

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Conditional independence Conditional independence: Joint probability:

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Marginal independence

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Marginal versus Conditional

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Marginal versus Conditional(cont.) Marginal conditional

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Homogeneous Association For a 2*2*K table, homogeneous XY association defined as: A symmetric property: – Applies to any pair of variables viewed across the categories of the third. – No interaction between two variables in their effects on the other variable.

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Homogeneous Association(cont.) Suppose: – X=smoking(yes, no) – Y=lung cancer(yes, no) – Z=age( 65) – And Age is an Effect Modifier

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Extensions for i*j Tables For a 2*2 table Odds ratio An i*j table Odds ratios

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Representation methods Method 1

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Method 2

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For I*J tables (I-1)*(J-1) odds ratios describe any association All 1.0s means INDEPENDENCE! Three-way I*J*K tables, Homogeneous XY association means: any conditional odds ratio formed using two categories of X and Y each is the same at each category of Z.

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Measures of Association Two kinds of variables: – Nominal variables – Ordinal variables Nominal variables: Set a measure for X and Y: – V(Y),V(Y|X) Proportional reduction:

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Measures of variation Entropy: Goodman and Kruskal(1954) (tau) Lambda:

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About Entropy Uncertainty coefficient: U=0=>INDEPENDENCE U=1=>π(j|i)=1 for each i, some j. Drawbacks: No intuition for such a proportional reduction.

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Ordinal Trends An example:

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Three kinds of relationship Concordant Discordant Tied

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Example(cont.) D=849 Define (C-D)/(C+D) as Gamma measure. Here, A weak tendency for job satisfaction to increase as income increases.

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Generalized

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Properties of Gamma Measure Symmetric Range [-1,1] Absolute value of 1 means perfect linear Monotonicity is required for Independence =>,not vice-versa.

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